[{"external_id":{"arxiv":["2310.06677"],"isi":["001385326500001"]},"intvolume":"        26","oa_version":"Published Version","day":"01","department":[{"_id":"LaEr"}],"year":"2025","isi":1,"type":"journal_article","publisher":"Springer Nature","ec_funded":1,"doi":"10.1007/s00023-024-01518-y","language":[{"iso":"eng"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","citation":{"mla":"Erdös, László, et al. “Prethermalization for Deformed Wigner Matrices.” <i>Annales Henri Poincare</i>, vol. 26, Springer Nature, 2025, pp. 1991–2033, doi:<a href=\"https://doi.org/10.1007/s00023-024-01518-y\">10.1007/s00023-024-01518-y</a>.","ieee":"L. Erdös, S. J. Henheik, J. Reker, and V. Riabov, “Prethermalization for deformed Wigner matrices,” <i>Annales Henri Poincare</i>, vol. 26. Springer Nature, pp. 1991–2033, 2025.","ama":"Erdös L, Henheik SJ, Reker J, Riabov V. Prethermalization for deformed Wigner matrices. <i>Annales Henri Poincare</i>. 2025;26:1991-2033. doi:<a href=\"https://doi.org/10.1007/s00023-024-01518-y\">10.1007/s00023-024-01518-y</a>","apa":"Erdös, L., Henheik, S. J., Reker, J., &#38; Riabov, V. (2025). Prethermalization for deformed Wigner matrices. <i>Annales Henri Poincare</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00023-024-01518-y\">https://doi.org/10.1007/s00023-024-01518-y</a>","chicago":"Erdös, László, Sven Joscha Henheik, Jana Reker, and Volodymyr Riabov. “Prethermalization for Deformed Wigner Matrices.” <i>Annales Henri Poincare</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00023-024-01518-y\">https://doi.org/10.1007/s00023-024-01518-y</a>.","short":"L. Erdös, S.J. Henheik, J. Reker, V. Riabov, Annales Henri Poincare 26 (2025) 1991–2033.","ista":"Erdös L, Henheik SJ, Reker J, Riabov V. 2025. Prethermalization for deformed Wigner matrices. Annales Henri Poincare. 26, 1991–2033."},"publication":"Annales Henri Poincare","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","ddc":["510"],"author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","full_name":"Erdös, László","first_name":"László","orcid":"0000-0001-5366-9603"},{"first_name":"Sven Joscha","orcid":"0000-0003-1106-327X","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","last_name":"Henheik","full_name":"Henheik, Sven Joscha"},{"first_name":"Jana","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","last_name":"Reker","full_name":"Reker, Jana"},{"first_name":"Volodymyr","full_name":"Riabov, Volodymyr","last_name":"Riabov","id":"1949f904-edfb-11eb-afb5-e2dfddabb93b"}],"OA_type":"hybrid","date_created":"2025-01-05T23:01:59Z","license":"https://creativecommons.org/licenses/by/4.0/","OA_place":"publisher","file":[{"date_updated":"2025-06-25T05:38:34Z","checksum":"49e6a934db540206f7eaa0c798553ded","file_size":977773,"success":1,"creator":"dernst","content_type":"application/pdf","access_level":"open_access","file_id":"19895","date_created":"2025-06-25T05:38:34Z","relation":"main_file","file_name":"2025_AnnalesHenriPoincare_Erdoes.pdf"}],"fulldoi":"https://doi.org/10.1007/s00023-024-01518-y","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"17174"},{"id":"20575","status":"public","relation":"dissertation_contains"},{"relation":"dissertation_contains","status":"public","id":"19540"}]},"acknowledgement":"All authors were supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.\r\nJ.R. was additionally supported by the ERC Advanced Grant “LDRaM” No. 884584.\r\nWe thank Peter Reimann and Lennart Dabelow for helpful comments. Open access funding provided by Institute of Science and Technology (IST Austria).","publication_identifier":{"issn":["1424-0637"]},"article_type":"original","abstract":[{"text":"We prove that a class of weakly perturbed Hamiltonians of the form H_λ= H_0 + λW, with W being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by H_λ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order λ^{-2}. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix H_λ.","lang":"eng"}],"_id":"18764","scopus_import":"1","date_published":"2025-06-01T00:00:00Z","quality_controlled":"1","corr_author":"1","date_updated":"2026-07-29T13:18:17Z","oa":1,"file_date_updated":"2025-06-25T05:38:34Z","page":"1991-2033","volume":26,"status":"public","title":"Prethermalization for deformed Wigner matrices","arxiv":1,"project":[{"grant_number":"101020331","call_identifier":"H2020","_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta"}],"month":"06"},{"main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2212.14638"}],"OA_type":"green","author":[{"orcid":"0000-0001-6892-8137","first_name":"Guillaume","full_name":"Dubach, Guillaume","id":"D5C6A458-10C4-11EA-ABF4-A4B43DDC885E","last_name":"Dubach"},{"full_name":"Reker, Jana","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","last_name":"Reker","first_name":"Jana"}],"date_created":"2024-05-23T08:31:57Z","OA_place":"repository","fulldoi":"https://doi.org/10.1142/s2010326324500072","related_material":{"record":[{"id":"17164","relation":"dissertation_contains","status":"public"}]},"article_type":"original","publication_identifier":{"eissn":["2010-3271"],"issn":["2010-3263"]},"article_number":"2450007","abstract":[{"lang":"eng","text":"We provide a dynamical study of a model of multiplicative perturbation of a unitary matrix introduced by Fyodorov. In particular, we identify a flow of deterministic domains that bound the spectrum with high probability, separating the outlier from the typical eigenvalues at all sub-critical timescales. These results are obtained under generic assumptions on U that hold for a variety of unitary random matrix models."}],"_id":"17047","scopus_import":"1","date_published":"2024-04-01T00:00:00Z","quality_controlled":"1","corr_author":"1","date_updated":"2026-04-07T13:02:12Z","oa":1,"volume":13,"status":"public","title":"Dynamics of a rank-one multiplicative perturbation of a unitary matrix","arxiv":1,"project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","call_identifier":"H2020","name":"Random matrices beyond Wigner-Dyson-Mehta"}],"month":"04","external_id":{"arxiv":["2212.14638"],"isi":["001229295200002"]},"intvolume":"        13","oa_version":"Preprint","day":"01","issue":"2","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"year":"2024","isi":1,"type":"journal_article","publisher":"World Scientific Publishing","ec_funded":1,"doi":"10.1142/s2010326324500072","publication_status":"published","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"citation":{"ista":"Dubach G, Reker J. 2024. Dynamics of a rank-one multiplicative perturbation of a unitary matrix. Random Matrices: Theory and Applications. 13(2), 2450007.","short":"G. Dubach, J. Reker, Random Matrices: Theory and Applications 13 (2024).","apa":"Dubach, G., &#38; Reker, J. (2024). Dynamics of a rank-one multiplicative perturbation of a unitary matrix. <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/s2010326324500072\">https://doi.org/10.1142/s2010326324500072</a>","chicago":"Dubach, Guillaume, and Jana Reker. “Dynamics of a Rank-One Multiplicative Perturbation of a Unitary Matrix.” <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing, 2024. <a href=\"https://doi.org/10.1142/s2010326324500072\">https://doi.org/10.1142/s2010326324500072</a>.","ieee":"G. Dubach and J. Reker, “Dynamics of a rank-one multiplicative perturbation of a unitary matrix,” <i>Random Matrices: Theory and Applications</i>, vol. 13, no. 2. World Scientific Publishing, 2024.","ama":"Dubach G, Reker J. Dynamics of a rank-one multiplicative perturbation of a unitary matrix. <i>Random Matrices: Theory and Applications</i>. 2024;13(2). doi:<a href=\"https://doi.org/10.1142/s2010326324500072\">10.1142/s2010326324500072</a>","mla":"Dubach, Guillaume, and Jana Reker. “Dynamics of a Rank-One Multiplicative Perturbation of a Unitary Matrix.” <i>Random Matrices: Theory and Applications</i>, vol. 13, no. 2, 2450007, World Scientific Publishing, 2024, doi:<a href=\"https://doi.org/10.1142/s2010326324500072\">10.1142/s2010326324500072</a>."},"publication":"Random Matrices: Theory and Applications","article_processing_charge":"No"},{"date_created":"2024-06-21T09:31:17Z","fulldoi":"https://doi.org/10.1007/s11040-024-09483-y","related_material":{"record":[{"status":"public","id":"17164","relation":"dissertation_contains"}]},"file":[{"access_level":"open_access","file_id":"17175","content_type":"application/pdf","creator":"cchlebak","date_created":"2024-06-26T11:26:42Z","file_name":"2024_MathPhysAnaGeo_Reker.pdf","relation":"main_file","date_updated":"2024-06-26T11:26:42Z","success":1,"checksum":"7d04318d66f765621bdcb648378d458e","file_size":1327596}],"article_type":"original","article_number":"10","publication_identifier":{"eissn":["1572-9656"],"issn":["1385-0172"]},"abstract":[{"text":"We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of Male et al. (Random Matrices Theory Appl. 11(2):2250015, 2022), showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem given in the recent companion paper (Reker in Preprint, arXiv:2204.03419, 2023). and thus allow identifying the fluctuation around the thermal value in certain thermalization problems.","lang":"eng"}],"ddc":["519"],"author":[{"first_name":"Jana","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","last_name":"Reker","full_name":"Reker, Jana"}],"oa":1,"file_date_updated":"2024-06-26T11:26:42Z","status":"public","volume":27,"title":"Fluctuation moments for regular functions of Wigner Matrices","arxiv":1,"project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"},{"call_identifier":"H2020","grant_number":"101020331","_id":"62796744-2b32-11ec-9570-940b20777f1d","name":"Random matrices beyond Wigner-Dyson-Mehta"}],"month":"06","_id":"17154","scopus_import":"1","date_published":"2024-06-20T00:00:00Z","quality_controlled":"1","date_updated":"2026-04-07T13:02:12Z","intvolume":"        27","oa_version":"Published Version","day":"20","external_id":{"arxiv":["2307.11029"],"isi":["001251464300001"]},"citation":{"ieee":"J. Reker, “Fluctuation moments for regular functions of Wigner Matrices,” <i>Mathematical Physics, Analysis and Geometry</i>, vol. 27, no. 3. Springer Nature, 2024.","ama":"Reker J. Fluctuation moments for regular functions of Wigner Matrices. <i>Mathematical Physics, Analysis and Geometry</i>. 2024;27(3). doi:<a href=\"https://doi.org/10.1007/s11040-024-09483-y\">10.1007/s11040-024-09483-y</a>","apa":"Reker, J. (2024). Fluctuation moments for regular functions of Wigner Matrices. <i>Mathematical Physics, Analysis and Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11040-024-09483-y\">https://doi.org/10.1007/s11040-024-09483-y</a>","short":"J. Reker, Mathematical Physics, Analysis and Geometry 27 (2024).","chicago":"Reker, Jana. “Fluctuation Moments for Regular Functions of Wigner Matrices.” <i>Mathematical Physics, Analysis and Geometry</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s11040-024-09483-y\">https://doi.org/10.1007/s11040-024-09483-y</a>.","mla":"Reker, Jana. “Fluctuation Moments for Regular Functions of Wigner Matrices.” <i>Mathematical Physics, Analysis and Geometry</i>, vol. 27, no. 3, 10, Springer Nature, 2024, doi:<a href=\"https://doi.org/10.1007/s11040-024-09483-y\">10.1007/s11040-024-09483-y</a>.","ista":"Reker J. 2024. Fluctuation moments for regular functions of Wigner Matrices. Mathematical Physics, Analysis and Geometry. 27(3), 10."},"publication":"Mathematical Physics, Analysis and Geometry","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","issue":"3","year":"2024","department":[{"_id":"LaEr"}],"isi":1,"type":"journal_article","publisher":"Springer Nature","ec_funded":1,"doi":"10.1007/s11040-024-09483-y","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","publication_status":"published","language":[{"iso":"eng"}]},{"keyword":["Random Matrices","Spectrum","Central Limit Theorem","Resolvent","Free Probability"],"citation":{"ista":"Reker J. 2024. Central limit theorems for random matrices: From resolvents to free probability. Institute of Science and Technology Austria.","mla":"Reker, Jana. <i>Central Limit Theorems for Random Matrices: From Resolvents to Free Probability</i>. Institute of Science and Technology Austria, 2024, doi:<a href=\"https://doi.org/10.15479/at:ista:17164\">10.15479/at:ista:17164</a>.","ieee":"J. Reker, “Central limit theorems for random matrices: From resolvents to free probability,” Institute of Science and Technology Austria, 2024.","ama":"Reker J. Central limit theorems for random matrices: From resolvents to free probability. 2024. doi:<a href=\"https://doi.org/10.15479/at:ista:17164\">10.15479/at:ista:17164</a>","apa":"Reker, J. (2024). <i>Central limit theorems for random matrices: From resolvents to free probability</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:17164\">https://doi.org/10.15479/at:ista:17164</a>","chicago":"Reker, Jana. “Central Limit Theorems for Random Matrices: From Resolvents to Free Probability.” Institute of Science and Technology Austria, 2024. <a href=\"https://doi.org/10.15479/at:ista:17164\">https://doi.org/10.15479/at:ista:17164</a>.","short":"J. Reker, Central Limit Theorems for Random Matrices: From Resolvents to Free Probability, Institute of Science and Technology Austria, 2024."},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","short":"CC BY-NC-SA (4.0)","image":"/images/cc_by_nc_sa.png","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)"},"article_processing_charge":"No","has_accepted_license":"1","year":"2024","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"type":"dissertation","doi":"10.15479/at:ista:17164","ec_funded":1,"publisher":"Institute of Science and Technology Austria","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","language":[{"iso":"eng"}],"publication_status":"published","alternative_title":["ISTA Thesis"],"supervisor":[{"orcid":"0000-0001-5366-9603","first_name":"László","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös"}],"day":"26","oa_version":"Published Version","page":"206","file_date_updated":"2024-06-26T12:44:53Z","oa":1,"title":"Central limit theorems for random matrices: From resolvents to free probability","status":"public","month":"06","project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","call_identifier":"H2020","grant_number":"101020331","name":"Random matrices beyond Wigner-Dyson-Mehta"}],"_id":"17164","date_published":"2024-06-26T00:00:00Z","date_updated":"2026-04-07T13:02:13Z","corr_author":"1","OA_place":"publisher","license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","date_created":"2024-06-24T11:23:29Z","fulldoi":"https://doi.org/10.15479/at:ista:17164","file":[{"checksum":"fb16d86e1f2753dc3a9e14d2bdfd84cd","file_size":2783027,"date_updated":"2024-06-26T12:44:53Z","date_created":"2024-06-26T12:39:36Z","relation":"main_file","file_name":"ISTA_Thesis_JReker.pdf","content_type":"application/pdf","creator":"jreker","access_level":"open_access","file_id":"17176"},{"date_updated":"2024-06-26T12:44:53Z","file_size":3054878,"checksum":"cb1e54009d47c1dcf5b866c4566fa27f","content_type":"application/zip","creator":"jreker","access_level":"closed","file_id":"17177","relation":"source_file","file_name":"ISTA_Thesis_JReker_SourceFiles.zip","date_created":"2024-06-26T12:39:42Z"}],"related_material":{"record":[{"status":"public","id":"17173","relation":"part_of_dissertation"},{"status":"public","id":"11135","relation":"part_of_dissertation"},{"id":"17047","relation":"part_of_dissertation","status":"public"},{"status":"public","id":"17154","relation":"part_of_dissertation"},{"status":"public","relation":"part_of_dissertation","id":"17174"}]},"abstract":[{"lang":"eng","text":"This thesis is structured into two parts. In the first part, we consider the random\r\nvariable X := Tr(f1(W)A1 . . . fk(W)Ak) where W is an N × N Hermitian Wigner matrix, k ∈ N, and we choose (possibly N-dependent) regular functions f1, . . . , fk as well as\r\nbounded deterministic matrices A1, . . . , Ak. In this context, we prove a functional central\r\nlimit theorem on macroscopic and mesoscopic scales, showing that the fluctuations of X\r\naround its expectation are Gaussian and that the limiting covariance structure is given\r\nby a deterministic recursion. We further give explicit error bounds in terms of the scaling\r\nof f1, . . . , fk and the number of traceless matrices among A1, . . . , Ak, thus extending\r\nthe results of Cipolloni, Erdős and Schröder [40] to products of arbitrary length k ≥ 2.\r\nAnalyzing the underlying combinatorics leads to a non-recursive formula for the variance\r\nof X as well as the covariance of X and Y := Tr(fk+1(W)Ak+1 . . . fk+ℓ(W)Ak+ℓ) of similar\r\nbuild. When restricted to polynomials, these formulas reproduce recent results of Male,\r\nMingo, Peché, and Speicher [107], showing that the underlying combinatorics of noncrossing partitions and annular non-crossing permutations continue to stay valid beyond\r\nthe setting of second-order free probability theory. As an application, we consider the\r\nfluctuation of Tr(eitW A1e\r\n−itW A2)/N around its thermal value Tr(A1) Tr(A2)/N2 when t\r\nis large and give an explicit formula for the variance.\r\nThe second part of the thesis collects three smaller projects focusing on different random\r\nmatrix models. In the first project, we show that a class of weakly perturbed Hamiltonians\r\nof the form Hλ = H0 + λW, where W is a Wigner matrix, exhibits prethermalization.\r\nThat is, the time evolution generated by Hλ relaxes to its ultimate thermal state via an\r\nintermediate prethermal state with a lifetime of order λ\r\n−2\r\n. As the main result, we obtain\r\na general relaxation formula, expressing the perturbed dynamics via the unperturbed\r\ndynamics and the ultimate thermal state. The proof relies on a two-resolvent global law\r\nfor the deformed Wigner matrix Hλ.\r\nThe second project focuses on correlated random matrices, more precisely on a correlated N × N Hermitian random matrix with a polynomially decaying metric correlation\r\nstructure. A trivial a priori bound shows that the operator norm of this model is stochastically dominated by √\r\nN. However, by calculating the trace of the moments of the matrix\r\nand using the summable decay of the cumulants, the norm estimate can be improved to a\r\nbound of order one.\r\nIn the third project, we consider a multiplicative perturbation of the form UA(t) where U\r\nis a unitary random matrix and A = diag(t, 1, ..., 1). This so-called UA model was\r\nfirst introduced by Fyodorov [73] for its applications in scattering theory. We give a\r\ngeneral description of the eigenvalue trajectories obtained by varying the parameter t and\r\nintroduce a flow of deterministic domains that separates the outlier resulting from the\r\nrank-one perturbation from the typical eigenvalues for all sub-critical timescales. The\r\nresults are obtained under generic assumptions on U that hold for various unitary random\r\nmatrices, including the circular unitary ensemble (CUE) in the original formulation of\r\nthe model."}],"publication_identifier":{"issn":["2663-337X"]},"ddc":["519"],"author":[{"first_name":"Jana","full_name":"Reker, Jana","last_name":"Reker","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9"}],"degree_awarded":"PhD"},{"has_accepted_license":"1","tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"article_processing_charge":"Yes","publication":"Electronic Journal of Probability","citation":{"mla":"Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.” <i>Electronic Journal of Probability</i>, vol. 29, 191, Institute of Mathematical Statistics, 2024, doi:<a href=\"https://doi.org/10.1214/24-EJP1247\">10.1214/24-EJP1247</a>.","chicago":"Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2024. <a href=\"https://doi.org/10.1214/24-EJP1247\">https://doi.org/10.1214/24-EJP1247</a>.","apa":"Reker, J. (2024). Multi-point functional central limit theorem for Wigner matrices. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/24-EJP1247\">https://doi.org/10.1214/24-EJP1247</a>","short":"J. Reker, Electronic Journal of Probability 29 (2024).","ieee":"J. Reker, “Multi-point functional central limit theorem for Wigner matrices,” <i>Electronic Journal of Probability</i>, vol. 29. Institute of Mathematical Statistics, 2024.","ama":"Reker J. Multi-point functional central limit theorem for Wigner matrices. <i>Electronic Journal of Probability</i>. 2024;29. doi:<a href=\"https://doi.org/10.1214/24-EJP1247\">10.1214/24-EJP1247</a>","ista":"Reker J. 2024. Multi-point functional central limit theorem for Wigner matrices. Electronic Journal of Probability. 29, 191."},"publication_status":"published","language":[{"iso":"eng"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.1214/24-EJP1247","ec_funded":1,"publisher":"Institute of Mathematical Statistics","type":"journal_article","isi":1,"year":"2024","department":[{"_id":"LaEr"}],"oa_version":"Published Version","day":"20","intvolume":"        29","external_id":{"isi":["001381599200001"],"arxiv":["2307.11028"]},"month":"12","project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","call_identifier":"H2020"}],"arxiv":1,"title":"Multi-point functional central limit theorem for Wigner matrices","volume":29,"status":"public","file_date_updated":"2025-01-08T08:44:03Z","oa":1,"date_updated":"2025-09-09T11:59:15Z","DOAJ_listed":"1","quality_controlled":"1","corr_author":"1","date_published":"2024-12-20T00:00:00Z","scopus_import":"1","_id":"18762","abstract":[{"text":"Consider the random variable $\\mathrm{Tr}( f_1(W)A_1\\dots f_k(W)A_k)$ where $W$ is an $N\\times N$ Hermitian Wigner matrix, $k\\in\\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\\dots, f_k$ as well as bounded deterministic matrices $A_1,\\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\\dots,f_k$ and the number of traceless matrices among $A_1,\\dots,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\\geq2$. As an application, we consider the fluctuation of $\\mathrm{Tr}(\\mathrm{e}^{\\mathrm{i} tW}A_1\\mathrm{e}^{-\\mathrm{i} tW}A_2)$ around its thermal value $\\mathrm{Tr}(A_1)\\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.","lang":"eng"}],"publication_identifier":{"eissn":["1083-6489"]},"article_type":"original","article_number":"191","acknowledgement":"I am very grateful to László Erdős for suggesting the topic and many valuable discussions during my work on the project. I would also like to thank the two anonymous referees for their careful reading of the manuscript and detailed feedback.\r\nPartially supported by ERC Advanced Grants “RMTBeyond” No. 101020331 and “LDRaM” No. 884584.","file":[{"date_created":"2025-01-08T08:44:03Z","relation":"main_file","file_name":"2024_ElectrJournProbability_Reker.pdf","content_type":"application/pdf","creator":"dernst","access_level":"open_access","file_id":"18773","checksum":"67178feaa8630a332599d3037a3fe70e","file_size":812428,"success":1,"date_updated":"2025-01-08T08:44:03Z"}],"fulldoi":"https://doi.org/10.1214/24-EJP1247","related_material":{"record":[{"relation":"earlier_version","id":"17173","status":"public"}]},"date_created":"2025-01-05T23:01:58Z","OA_place":"publisher","OA_type":"gold","author":[{"full_name":"Reker, Jana","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","last_name":"Reker","first_name":"Jana"}],"ddc":["510"]},{"department":[{"_id":"LaEr"}],"year":"2023","_id":"17173","type":"preprint","date_published":"2023-07-21T00:00:00Z","doi":"10.48550/arXiv.2307.11028","date_updated":"2026-04-07T13:02:12Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"draft","language":[{"iso":"eng"}],"oa":1,"title":"Multi-point functional central limit theorem for Wigner Matrices","status":"public","citation":{"ista":"Reker J. Multi-point functional central limit theorem for Wigner Matrices. arXiv, 2307.11028.","short":"J. Reker, ArXiv (n.d.).","apa":"Reker, J. (n.d.). Multi-point functional central limit theorem for Wigner Matrices. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2307.11028\">https://doi.org/10.48550/arXiv.2307.11028</a>","chicago":"Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2307.11028\">https://doi.org/10.48550/arXiv.2307.11028</a>.","ama":"Reker J. Multi-point functional central limit theorem for Wigner Matrices. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2307.11028\">10.48550/arXiv.2307.11028</a>","ieee":"J. Reker, “Multi-point functional central limit theorem for Wigner Matrices,” <i>arXiv</i>. .","mla":"Reker, Jana. “Multi-Point Functional Central Limit Theorem for Wigner Matrices.” <i>ArXiv</i>, 2307.11028, doi:<a href=\"https://doi.org/10.48550/arXiv.2307.11028\">10.48550/arXiv.2307.11028</a>."},"article_processing_charge":"No","publication":"arXiv","arxiv":1,"month":"07","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2307.11028","open_access":"1"}],"author":[{"first_name":"Jana","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","last_name":"Reker","full_name":"Reker, Jana"}],"external_id":{"arxiv":["2307.11028"]},"date_created":"2024-06-26T08:54:56Z","OA_place":"repository","fulldoi":"https://doi.org/10.48550/arXiv.2307.11028","related_material":{"record":[{"id":"18762","status":"public","relation":"later_version"},{"status":"public","id":"17164","relation":"dissertation_contains"}]},"abstract":[{"lang":"eng","text":"Consider the random variable $\\mathrm{Tr}( f_1(W)A_1\\dots f_k(W)A_k)$ where $W$ is an $N\\times N$ Hermitian Wigner matrix, $k\\in\\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\\dots, f_k$ as well as bounded deterministic matrices $A_1,\\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\\dots,f_k$ and the number of traceless matrices among $A_1,\\dots,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\\geq2$. As an application, we consider the fluctuation of $\\mathrm{Tr}(\\mathrm{e}^{\\mathrm{i} tW}A_1\\mathrm{e}^{-\\mathrm{i} tW}A_2)$ around its thermal value $\\mathrm{Tr}(A_1)\\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance."}],"oa_version":"Preprint","article_number":"2307.11028","day":"21"},{"article_processing_charge":"No","publication":"arXiv","citation":{"ista":"Erdös L, Henheik SJ, Reker J, Riabov V. Prethermalization for deformed Wigner Matrices. arXiv, 2310.06677.","mla":"Erdös, László, et al. “Prethermalization for Deformed Wigner Matrices.” <i>ArXiv</i>, 2310.06677, doi:<a href=\"https://doi.org/10.48550/arXiv.2310.06677\">10.48550/arXiv.2310.06677</a>.","ama":"Erdös L, Henheik SJ, Reker J, Riabov V. Prethermalization for deformed Wigner Matrices. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2310.06677\">10.48550/arXiv.2310.06677</a>","ieee":"L. Erdös, S. J. Henheik, J. Reker, and V. Riabov, “Prethermalization for deformed Wigner Matrices,” <i>arXiv</i>. .","apa":"Erdös, L., Henheik, S. J., Reker, J., &#38; Riabov, V. (n.d.). Prethermalization for deformed Wigner Matrices. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2310.06677\">https://doi.org/10.48550/arXiv.2310.06677</a>","chicago":"Erdös, László, Sven Joscha Henheik, Jana Reker, and Volodymyr Riabov. “Prethermalization for Deformed Wigner Matrices.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2310.06677\">https://doi.org/10.48550/arXiv.2310.06677</a>.","short":"L. Erdös, S.J. Henheik, J. Reker, V. Riabov, ArXiv (n.d.)."},"ec_funded":1,"doi":"10.48550/arXiv.2310.06677","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","language":[{"iso":"eng"}],"publication_status":"draft","department":[{"_id":"LaEr"}],"year":"2023","type":"preprint","oa_version":"Preprint","day":"23","external_id":{"arxiv":["2310.06677"]},"arxiv":1,"month":"12","project":[{"name":"Random matrices beyond Wigner-Dyson-Mehta","_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","call_identifier":"H2020"}],"oa":1,"title":"Prethermalization for deformed Wigner Matrices","status":"public","date_updated":"2026-04-07T13:02:12Z","corr_author":"1","_id":"17174","date_published":"2023-12-23T00:00:00Z","fulldoi":"https://doi.org/10.48550/arXiv.2310.06677","related_material":{"record":[{"id":"18764","relation":"later_version","status":"public"},{"status":"public","relation":"dissertation_contains","id":"20575"},{"relation":"dissertation_contains","id":"17164","status":"public"}]},"abstract":[{"lang":"eng","text":"We prove that a class of weakly perturbed Hamiltonians of the form $H_λ= H_0 + λW$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by $H_λ$ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order $λ^{-2}$. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix $H_λ$."}],"article_number":"2310.06677","date_created":"2024-06-26T08:56:52Z","OA_place":"repository","author":[{"full_name":"Erdös, László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-5366-9603","first_name":"László"},{"orcid":"0000-0003-1106-327X","first_name":"Sven Joscha","full_name":"Henheik, Sven Joscha","id":"31d731d7-d235-11ea-ad11-b50331c8d7fb","last_name":"Henheik"},{"first_name":"Jana","full_name":"Reker, Jana","last_name":"Reker","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9"},{"last_name":"Riabov","id":"1949f904-edfb-11eb-afb5-e2dfddabb93b","full_name":"Riabov, Volodymyr","first_name":"Volodymyr"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2310.06677"}]},{"isi":1,"type":"journal_article","year":"2022","issue":"4","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"publisher":"World Scientific Publishing","doi":"10.1142/s2010326322500368","citation":{"mla":"Reker, Jana. “On the Operator Norm of a Hermitian Random Matrix with Correlated Entries.” <i>Random Matrices: Theory and Applications</i>, vol. 11, no. 4, 2250036, World Scientific Publishing, 2022, doi:<a href=\"https://doi.org/10.1142/s2010326322500368\">10.1142/s2010326322500368</a>.","ieee":"J. Reker, “On the operator norm of a Hermitian random matrix with correlated entries,” <i>Random Matrices: Theory and Applications</i>, vol. 11, no. 4. World Scientific Publishing, 2022.","ama":"Reker J. On the operator norm of a Hermitian random matrix with correlated entries. <i>Random Matrices: Theory and Applications</i>. 2022;11(4). doi:<a href=\"https://doi.org/10.1142/s2010326322500368\">10.1142/s2010326322500368</a>","short":"J. Reker, Random Matrices: Theory and Applications 11 (2022).","apa":"Reker, J. (2022). On the operator norm of a Hermitian random matrix with correlated entries. <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/s2010326322500368\">https://doi.org/10.1142/s2010326322500368</a>","chicago":"Reker, Jana. “On the Operator Norm of a Hermitian Random Matrix with Correlated Entries.” <i>Random Matrices: Theory and Applications</i>. World Scientific Publishing, 2022. <a href=\"https://doi.org/10.1142/s2010326322500368\">https://doi.org/10.1142/s2010326322500368</a>.","ista":"Reker J. 2022. On the operator norm of a Hermitian random matrix with correlated entries. Random Matrices: Theory and Applications. 11(4), 2250036."},"keyword":["Discrete Mathematics and Combinatorics","Statistics","Probability and Uncertainty","Statistics and Probability","Algebra and Number Theory"],"publication":"Random Matrices: Theory and Applications","article_processing_charge":"No","external_id":{"isi":["000848873800001"],"arxiv":["2103.03906"]},"intvolume":"        11","oa_version":"Preprint","day":"01","scopus_import":"1","date_published":"2022-10-01T00:00:00Z","_id":"11135","corr_author":"1","quality_controlled":"1","date_updated":"2026-04-07T13:02:12Z","status":"public","volume":11,"title":"On the operator norm of a Hermitian random matrix with correlated entries","oa":1,"month":"10","arxiv":1,"main_file_link":[{"url":" https://doi.org/10.48550/arXiv.2103.03906","open_access":"1"}],"author":[{"first_name":"Jana","last_name":"Reker","id":"e796e4f9-dc8d-11ea-abe3-97e26a0323e9","full_name":"Reker, Jana"}],"date_created":"2022-04-08T07:11:12Z","publication_identifier":{"issn":["2010-3263"],"eissn":["2010-3271"]},"article_number":"2250036","article_type":"original","abstract":[{"lang":"eng","text":"We consider a correlated NxN Hermitian random matrix with a polynomially decaying metric correlation structure. By calculating the trace of the moments of the matrix and using the summable decay of the cumulants, we show that its operator norm is stochastically dominated by one."}],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"17164"}]},"fulldoi":"https://doi.org/10.1142/s2010326322500368"}]